Abstract
We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.
Highlights
Introduction and preliminaries LetA be an algebra
Singer and Wermer [ ] obtained a fundamental result which started the investigation of the ranges of linear derivations on Banach algebras
Considering the base of the previous result, we show that every approximate ring left derivation on a semiprime normed algebra maps into its center and by using this fact, we prove that every approximate linear left derivation on a semisimple Banach algebra is continuous
Summary
Since semisimple algebras are semiprime [ ], Theorem . Let A be a semiprime unital Banach algebra. ), it is reduced to the equation δ(xyx) = x δ(y) + xyδ(x) – yxδ(x) for all x, y ∈ A. From Vukman’s result [ ], we see that δ is a linear derivation with δ(A) ⊆ Z(A). Let A be a unital semisimple Banach algebra. We consider the result which is needed in the following theorems. If A is semiprime or unital, ξ and η are linear mappings. Let A be a semiprime Banach algebra. It follows from the result in [ ] that there exists a unique additive mapping D : A → A defined by δ(snx).
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